Thermodynamics - compression of a gas in a piston-cylinder device

[roldy]

Homework Statement

An ideal gas initially at 100kPa and 300K is contained in a piston-cylinder device with an initial volume of 5 liters. The gas is then compressed slowly according to the relationship that pV^2=constant until a final temperature of 400K is reached. Sketch the process on a p-V diagram and find how much work is done in this process.

Homework Equations

pV^2=costant

The Attempt at a Solution

I'm not sure in my logic but since this is a freely moving piston-cylinder device, the work should be 0 which would result in a horizonatal line. Is this assumption correct?

 

[chemisttree]

No. Review the definition of work as it pertains to pressure and volume changes.

 

[Blueshift5]

Thank you for your question. I can provide a response to this content.

Firstly, your assumption that the work done is 0 is not entirely correct. While it is true that there is no external work done on the gas, there is still work being done by the gas itself as it expands against the piston.

To better understand this process, we can use the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. In this case, since the gas is being compressed, the work done by the gas will be negative (since work done on a system is considered positive).

To sketch the process on a p-V diagram, we can plot the initial state of the gas (100kPa, 300K) as a point on the graph and then use the relationship pV^2=constant to plot the path of the gas as it is compressed. This will result in a curve that decreases in volume as pressure increases.

To calculate the work done, we can use the formula W = -∫p dV, where the integral is taken over the path of the gas on the p-V diagram. This will give us the area under the curve, which represents the work done by the gas. In this case, since the process is isentropic (constant entropy), the work done can also be calculated using the formula W = nRTln(V2/V1), where n is the number of moles of gas, R is the gas constant, and V2 and V1 are the final and initial volumes, respectively.

I hope this helps to clarify the process and calculations involved in the compression of a gas in a piston-cylinder device. If you have any further questions, please let me know.